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```ON  THE MOTION  OF  PENDULUMS.
135
+ (2nt — I)"1 to the difference of the logarithms, we may apply it to the deduced value of VX, which is proportional to the difference of the logarithms. In Coulomb's experiments 10 oscillations were observed, and therefore 2nt = 20?r, and (%nt-~ I)"1 = 0'01617, and the uncorrected value of VA6' being 0*0555, we get 0*0009 for the correction, giving V/^ = 0*0564.
NOTE 0.   Article 50.
The results mentioned in this article were originally given without demonstration; but as the mode in which they were obtained is short, and by no means obvious, I have thought it advisable to add the demonstrations.
In order that the right-hand members of equations (138) may be perfect differentials, we must have
d\$ dy
dz
dy '
dm'"
-<>
~  '
dS    da>'    .
"7—r   7~ == v,
dz    dy
d\$    dw"'___ dx     dy ~"   ;
dco"'    dco'    A
-—=------1—=— = 0,
dz      dx
dS     dco"
dx      dz
dS     dco'
dy     dz
da)'     dco"
dx       dy
The equations (c) give da
dx
= 0,
f-=0,
dm"
dz
= 0.
.(d).
In the particular case in which S = 0, the equations (a), (6), and
(d) give
da = 0,        do)" = 0,       cZa>"' = 0,
and therefore co, w", and «'" are constant as stated in Art. 50. In the general case the equations (a), (6), and (d) give for the differentials of o>', a>", and a/" the following expressions:
da' =— 1  dy + -y-
d^    J       cZv/
^
7       , dz
d\$ j +   ,- dy u
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